#### Quadrangulations: Constructions and Geodesics

##### Wednesday, Jan 29, 2020

We previously worked with general two-complexes, including those representing surfaces. We now specialize to the first of the three special representations that are most useful to us - *non-positive quadrangulations*. Here we discuss what these are, how to construct these and the geodesics for these.

##### Non-positive quadrangulations

For a closed surface `$\Sigma$`

, a non-positive quadrangulation is a two-complex homeomorphic to `$\Sigma$`

such that

- Every face has four sides.
- The degree of (i.e., number of edges terminating in) each vertex is at least
`$5$`

.

##### Constructing quadrangulations

Start with a `$2$`

-complex `$\Delta$`

representing `$\Sigma$`

. We describe a general procedure for constructing quadrangulations `$Q$`

. If the `$2$`

-complex `$\Delta$`

has one vertex and one face, and the surface has genus at least `$2$`

, then the quadrangulation `$Q$`

we construct is automatically negatively curved.

The quadrangulation is the two complex `$Q$`

given in terms of `$\Delta$`

as follows.

**Vertices:**the vertices of`$Q$`

are the vertices of`$\Delta$`

together with a vertex`$b_f$`

for each face`$f$`

of`$\Delta$`

(thought of as the*barycenter*of`$f$`

).**Edges:**For a face`$f$`

with`$n$`

sides, with boundary`$e_0(f), e_1(f), \dots, e_{n-1}(f)$`

, we have`$n$`

edges`$\theta_i(f)$`

,`$0 \leq i < n$`

, with`$\theta_i$`

starting at`$b_f$`

and ending at`$e_i(f)$`

. Note that the edges of`$\Delta$`

are not edges of`$Q$`

.**Faces:**There is one face`$\Phi(\eta)$`

in`$Q$`

for each edge-pair`$(\eta, \bar\eta)$`

, where*bar*denotes flip. Namely, we have unique faces`$f$`

and`$g$`

and indices`$i$`

and`$j$`

so that`$\eta=e_i(f)$`

and`$\bar\eta = e_j(g)$`

. The corresponding face`$\Phi(\eta)$`

is the quadrilateral with boundary`$\theta_i(f)$`

,`$\bar\theta_{j + 1}(g)$`

,`$\theta_j (g)$`

,`$\bar\theta_i(f)$`

(check this before implementing).

##### Geodesics

Fix a non-positive quadrangulation, so each vertex has degree at least `$5$`

, and let `$\eta$`

be an edge. The left edge `$\theta$`

following `$\eta$`

is the successor in the unique face containing `$\eta$`

as a directed edge. This is the next edge in the sharpest left turn. We call going from `$\eta$`

to `$\theta$`

a *left turn*.

The *slight left turn* is the next sharpest left turn, obtained by taking a left turn, flipping and taking a left turn again (remember that this is a 5 or more pointed junction, so the terminology is justified). We can similarly define *right turn* and *slight right turn* (using flips and predecessors) as indicated in the previous note.

If an edge path has a segment where we make a left turn, followed by `$0$`

or more slight left turns and then another left turn, it is not a geodesic, and similarly with rights in place of lefts. Indeed in both these cases there is a clear shortening transformation that keeps the ends fixed and is a homotopy. We can apply such transformations till there are no obvious ones left.

The key point about non-positive curvature is that if there are no shortenings of the above form, the curve we have is a *geodesic*. Further geodesics are almost unique, and there is indeed a canonical leftmost geodesics. Many questions can be addressed using these.